Question

If \( y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \), then \[ \frac{dy}{dx} = \]

• A. \( y - 1 \)
• B. \( y + 1 \)
• C. \( y^2 - 1 \)
• D. \( y \)

Hint:

For series expansions like \( e^x \), the derivative is the same as the function itself.

Answer 1

The Correct Option is D

Step 1: Recognize the function.
The given function is the series expansion of \( e^x \). Thus, we can express the function as: \[ y = e^x \] Step 2: Differentiate.
Differentiating \( y = e^x \) gives: \[ \frac{dy}{dx} = e^x = y \] Step 3: Conclusion.
The correct answer is (D) \( y \).